Measurements of Large Distance

Large astronomical distances — such as the distance of a planet or star from the Earth — cannot be measured directly with anything like a metre scale, so astronomers instead use the parallax method.

To understand parallax, try a simple experiment: hold your hand out in front of you, and look at it first with your left eye closed, then with your right eye closed. Your hand will appear to shift position against the more distant background, purely because you viewed it from two slightly different points (your two eyes). This apparent shift in an object's position, caused only by a change in the position of the observer, is called parallax. If you know the separation between the two viewing points ($E_1E_2$, the distance between your eyes) and you measure the parallax angle $\theta$ through which the object appeared to shift, you can work out the distance $OP$ of the object as $OP = E_1E_2/\theta$.

For astronomical objects, the distances involved are far too large for a two-eye baseline to produce any measurable shift, so astronomers instead choose two widely separated points, A and B, on the surface of the Earth, separated by a known straight-line baseline distance $b$. Observers at A and B simultaneously sight a distant planet S, and measure the angle $\angle ASB$ between their two lines of sight — this angle, again called $\theta$, is the parallax angle. Because the planet's distance $D$ from Earth is enormous compared to the baseline $b$ (so that $b/D \ll 1$), the parallax angle $\theta$ is very small, and the arc $AB$ (of length $b$) can be treated as the arc of a circle of radius $D$ centred at the planet. This gives the small-angle relation …